Ruelle zeta function

In mathematics, the Ruelle zeta function is a zeta function associated with a dynamical system.

Formal definition

Let f be a function defined on a manifold M, such that the set of fixed points Fix(f ⁿ) is finite for all n > 1. Further let φ be a function on M with values in d × d complex matrices. The zeta function of the first kind is^[1]

Examples

In the special case d = 1, φ = 1, we have^[1]

which is the Artin–Mazur zeta function.

The Ihara zeta function is an example of a Ruelle zeta function.^[2]

See also

• List of zeta functions

References

• Lapidus, Michel L.; van Frankenhuijsen, Machiel (2006). Fractal geometry, complex dimensions and zeta functions. Geometry and spectra of fractal strings. Springer Monographs in Mathematics. New York, NY: Springer-Verlag. ISBN 0-387-33285-5. Zbl 1119.28005. 
• Kotani, Motoko; Sunada, Toshikazu (2000). "Zeta functions of finite graphs". J. Math. Sci. Univ. Tokyo 7: 7–25. 

• Terras, Audrey (2010). Zeta Functions of Graphs: A Stroll through the Garden. Cambridge Studies in Advanced Mathematics 128. Cambridge University Press. ISBN 0-521-11367-9. Zbl 1206.05003.